Closure-characteristic subgroup: Difference between revisions

Revision as of 19:12, 20 December 2014

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties [SHOW MORE]
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Definition

QUICK PHRASES: normal closure is characteristic, join of all conjugates is characteristic

A subgroup of a group is termed closure-characteristic, or a subgroup whose normal closure is characteristic, if its normal closure in the whole group is a characteristic subgroup. In symbols, a subgroup $H$ of a group $G$ is termed closure-characteristic if the normal closure $H^{G}$ of $H$ in $G$ is characteristic in $G$ .

Relation with other properties

Stronger properties

Property	Meaning	Intermediate notions
Characteristic subgroup	invariant under all automorphisms	\|FULL LIST, MORE INFO
Automorph-conjugate subgroup	all automorphic subgroups are conjugate	\|FULL LIST, MORE INFO
Join of automorph-conjugate subgroups	join of automorph-conjugate subgroups	\|FULL LIST, MORE INFO
Sylow subgroup	$p$ -subgroup of finite group with index relatively prime to $p$	\|FULL LIST, MORE INFO
Hall subgroup	subgroup of finite group whose order and index are relatively prime	\|FULL LIST, MORE INFO
Contranormal subgroup	normal closure is whole group	\|FULL LIST, MORE INFO

Conjunction with other properties

Any normal subgroup that is also closure-characteristic, is characteristic.

Metaproperties

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

Join-closedness

YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are join-closed | View variations of this property that are not join-closed
ABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness

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 {{quick phrase|[[quick phrase::normal closure is characteristic]], [[quick phrase::join of all conjugates is characteristic]]}}
-===Symbol-free definition===
-A [[subgroup]] of a [[group]] is termed '''closure-characteristic''' if its [[defining ingredient::normal closure]] in the whole group is a [[defining ingredient::characteristic subgroup]].
+A [[subgroup]] of a [[group]] is termed '''closure-characteristic''', or a '''subgroup whose normal closure is characteristic''', if its [[defining ingredient::normal closure]] in the whole group is a [[defining ingredient::characteristic subgroup]]. In symbols, a [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''closure-characteristic''' if the [[normal closure]] <math>H^G</math> of <math>H</math> in <math>G</math> is [[characteristic subgroup|characteristic]] in <math>G</math>.
-===Definition with symbols===
-A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''closure-characteristic''' if the [[normal closure]] <math>H^G</math> of <math>H</math> in <math>G</math> is [[characteristic subgroup|characteristic]] in <math>G</math>.
 ==Relation with other properties==